7.1 Pythagorean Theorem and 2D Applications The Pythagorean Theorem states that IF a triangle is a right triangle, THEN the sum of the squares of the lengths of the legs equals the square of the hypotenuse lengths. That’s a complicated way to say that if the legs of the triangle measure and and the hypotenuse measures , then . While you may have heard this in the past, we will now prove it.
Proof of the Pythagorean Theorem There are many ways to prove the Pythagorean Theorem, but take a look at the following picture. We will refer to this for our proof.
a
b c
b
c
a
In this picture we have a large square whose side lengths are equal to and an inner square whose side lengths are . Notice that if we find the area of the large square and subtract the area of the triangles we get the area of the inner square. So let’s do that algebraically.
a
c
c
b
The area of the larger square is:
b
2
a
and since there are four of them, the total area of the triangles is 2
The area of each triangle is
The area of the inner square is .
This means the large square minus the triangles would look like this: 2
Notice that the 2 and the 2 . which is that
2
.
cancel each other out (become zero), so we do get the result we expect
Do a search online to see if you can find another proof for this vital theorem. One more time, the IF‐THEN statement for the Pythagorean Theorem is: IF it’s a right triangle, THEN
must be true, we can now solve for any Since we know that in a right triangle the statement missing side length given the other two side lengths. The process of solving for a missing leg ( or ) is only slightly different from solving for a missing hypotenuse ( ).
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Solving for a Missing Leg Let’s first solve for a missing leg. First note that it makes no difference which leg we label as and which leg we label as . This is because the commutative property says that we can add in any order. In other words, or doesn’t matter, it will always equal . So if we are missing the length of a whether we have leg, it might be easiest to always assume it is that is missing.
12 in.
13 in.
Given the fact that this is a right triangle, we can solve for the missing leg length, . Just substitute everything we know into the Pythagorean Formula. We know that the hypotenuse length, , is 13 inches and that the other leg length, , is 12 inches. 12
a
13
Now go ahead and multiple out those exponents to get the following statement: 144
169
Notice this is a two‐step equation where is being squared and then increased by 144. Applying inverse operations, we know we should subtract 144 from both sides and then take the square root. That looks like this: 144 144
169 144 25 √25 5
We have just proved that the missing side length must be 5 inches.
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Sometimes the missing side length will be labeled with a different variable just to throw us off. Just remember that the legs are always and in the Pythagorean Formula and that , or the hypotenuse, is always the longest side length. For example, in the following picture which are the legs and which is the hypotenuse? The hypotenuse is always opposite (or across from) the right angle and is the longest side. So the hypotenuse in this picture is 10 ft. That means that the 6 ft and the must be the two sides. Notice that the legs can also be identified by the fact that they are the sides that make up the right angle. Now substitute into the Pythagorean Formula to solve for .
10 ft. x
6 ft.
6
10
36
100
36
36
So we know that the missing side length is 8 ft. in this particular triangle.
64 √
√64 8
Solving for a Missing Hypotenuse Let’s now solve for a missing hypotenuse. Remember that the hypotenuse is always the longest side and the side opposite the right angle. Take a look at this example.
15 ft.
8 ft.
Note that 8 ft. and 15 ft. must the lengths of the legs since they make up the right angle. That means that in this case is the missing hypotenuse. Plugging those values into the Pythagorean Formula yields the following: 8
x
64
15 225
Be careful at this point. Many students mistakenly try to subtract either 64 or 225 from both sides, but that is not accurate. We always combine like terms before using inverse operations, and in this case we still need to combine the 64 + 225 to get 289. So our next steps should look like this: 289
√289 17
This means that the missing hypotenuse length is 17 feet. Note that the only inverse operation we needed to apply in this case was the square root. 294
Let’s look at one more example of solving for a missing hypotenuse. Consider the following picture. Note that is the hypotenuse in this case because the sides with lengths 3 and 4 make up the right angle. Plug these values into the Pythagorean Formula.
3 cm.
3
y
4 9
16
25
4 cm.
√25
5
So the hypotenuse has a length of 5 centimeters in this case.
Pythagorean Theorem Word Problems The use of the Pythagorean Theorem can applied to word problems just as easily. For example, if we know that it is 90 feet from home plate to first base and 90 feet from first base to second base, how far would the catcher have to throw the baseball to get a runner out who is stealing second base? The best tip to give for solving word problems like this is to draw a picture. In this case, note that the distance from second base to home plate is the hypotenuse of the triangle. That means that the 90 feet distances are the legs. We can now solve as follows. 90 8100
90
8100
16200
√16200 127.3
For this problem, there was no exact square root. That means that
√16200 is irrational and it’s probably best to estimate this number. Our answer is approximated to the nearest one decimal place giving us about 127.3 feet. So the catcher would have to throw just over 127 feet to get out the runner trying to steal second base.
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Lesson 7.1 Find the missing side length of each right triangle. Round your answers to three decimal places if necessary. 1. 2. 3. 29 20 40 4 3 9 4. 5. 6. 73 25 55 5 10 3 7. 8. 9. 40 28 15 30 45 8
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10. 26 10
11. 12
12. 61
13 60
13. 14. 15. 45 20 28 40 7 21 Solve the following problems. Round your answers to the nearest whole number when necessary. 16. You’re locked out of your house, and the only open window is on the second floor 25 feet above the ground. There are bushes along the side of the house that force you to put the base of the ladder 7 feet away from the base of the house. How long of a ladder will you need to reach the window? 17. Shae takes off from her house and runs 3 miles north and 4 miles west. Tired, she wants to take the shortest route back. How much farther will she have to run if she heads straight back to her house? 297
18. Televisions are advertised by the length of their diagonals. If a 42 inch television measures 18 inches high, how wide is the television? 19. A soccer field is 100 yards by 60 yards. How long is the diagonal of the field? 20. You place a 24 foot ladder 10 feet away from the house. The top of the ladder just reaches a window on the second floor. How high off the ground is the window? 21. A rectangular garden measures 5 feet wide by 12 feet long. If a hose costs $5 per foot, how much would it cost to place a hose through the diagonal of the garden? 22. A rectangular dog pen is 3 meters by 4 meters. If a chain costs $1.75 per meter, how much would it cost to put a chain along the diagonal of the pen? 23. A rectangular park measures 8 miles long by 6 miles wide. The park director wants to put a fence along both sides of the trail that runs diagonally through the park. If the fence costs $150 per mile, how much will it cost to buy the fence? 24. A rectangular pool has a diagonal of 17 yards and a length of 15 yards. If the paint costs $2 per yard of coverage, how much will it cost the owner to paint the width of both ends of the pool? 298
